Concentration dependence of resistance components in solutions containing dissolved Fe2+/Fe3+

Electrolyte solutions containing Fe2+/Fe3+ are suitable for liquid thermoelectric conversion devices (LTEs), because they are inexpensive materials and exhibit a high electrochemical Seebeck coefficient α. Here, we investigated the concentration (c) dependence of resistance components, i.e., solvent (Rs), charge-transfer (Rct), and diffusion (Rdif) resistances, of dissolved-Fe2+/Fe3+-containing aqueous, methanol (MeOH), acetone, and propylene carbonate (PC) solutions. We found that the c dependence of Rs and Rdif are well reproduced by empirical formulas, and , where η(c) is viscosity at c. We further found that the magnitudes of Cs and Cdif are nearly independent of solvent, suggesting that η is one of the significant solution parameters that determine Rs and Rdif.


Introduction
Energy-harvesting devices are attracting the current attention of researchers from the viewpoint of a basic power source for the internet of things (IoT) society as well as sustainable development goals (SDGs).Thermoelectric conversion devices (TEs) are promising because they cover the eld from bioelectric and wearable electronics to industrial power generation.In particular, exible TEs using cellulose gel 1 have excellent compatibility with various thermoelectric conversion materials.This is because cellulose can be used in the design of exible and ion/ electron conductive materials with robust mechanical properties.Flexible TEs are expected to be used not only for thermoelectric conversion but also for sensors and refrigeration units.
Among TEs, liquid thermoelectric conversion devices (LTEs) are promising because they are made of inexpensive materials.There is already a long history of LTE research. 2][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] The performance of LTEs is governed by the electrochemical Seebeck coefficient a, effective electric conductivity s, and effective thermal conductivity k of the electrolyte. 20Unlike solid thermoelectric devices, s is related to the charge transfer and diffusion processes of redox ions as well as the conventional ion migration.The magnitude of s depends on the microscopic structure and material of the electrodes. 16,17In addition, effective s and k are inuenced by convection of the electrolyte induced by DT.
The dimensionless gure of merit (ZT ¼ a 2 sT k , where T is temperature) is a measure of the LTE performance.With the increase of ZT, the thermal efficiency h increases toward the Carnot efficiency , which is the maximum efficiency of a heat engine. 21To enhance ZT, it is effective to increase (decrease) a and s (k).
][6][7][8][9][10][11][12][13][14][15][16][17] This is because the organic electrolytes exhibit both large a and small k.In several organic solutions containing Fe 2+ /Fe 3+ , a is higher than the value (= 1.4 mV K −1 ) of an aqueous solution.For example, a is 3.6 mV K −1 in acetone solution and 1.8 mV K −1 in propylene carbonate (PC) solution. 22In addition, k of a typical organic solvent is z 0.2 W K −1 m −1 and is approximately 33% of the value (= 0.6 W K −1 m −1 ) of water.Recently, Wake et al. 18,19 showed that LTEs composed of dissolved-Fe 2+ /Fe 3+ -containing methanol (MeOH) and acetone solutions exhibit a large power factor (PF = a 2 s) comparable to that of the corresponding aqueous LTE.They also reported a and s against solute concentration c.The disadvantage of organic electrolytes is a small s value compared with that of an aqueous electrolyte.Except for aqueous electrolyte, 7 there exists no detailed investigation on the resistance components.Therefore, the origin of the small s in organic electrolytes is still unclear.Here, we will investigate the resistance components of several solutions containing dissolved Fe 2+ /Fe 3+ against c to deeply understand s and to obtain guidelines for increasing s in organic electrolytes.
In general, the resistance R (¼ where i 0 is the exchange current.Thus, R ct is proportional to k B T i 0 e and is independent of d.The physical meaning of R dif is as follows.As the reaction progresses, the concentration of reactants/products at the electrode surface changes in a way that prevents further reaction.For the reaction to continue, the reactants/products must diffuse into/from the bulk region.Note that the diffusion current of reactants/ products is driven by the concentration gradient created by the reaction at the electrode surface and is independent of d.
In this work, we investigated the c-dependence of R s , R ct , and R dif of dissolved-Fe 2+ /Fe 3+ -containing aqueous, MeOH, acetone, and PC solutions.We found that the c-dependence of R s and R dif is well reproduced by empirical formulas, R s À1 ¼ C s c hðcÞ and We further found that their coefficients, C s and C dif , are nearly independent of solvent, suggesting that h is one of the signicant solution parameters that determine R s and R dif .

Solution preparation
In this study, water, MeOH, acetone, and PC were selected as the solvents because they exhibit a high solubility of Fe(ClO 4 ) 2 / Fe(ClO 4 ) 3 .We prepared aqueous, MeOH, acetone, and PC solutions containing c M Fe(ClO 4 ) 2 $6.0H 2 O and c M Fe(ClO 4 ) 3 -$7.1H 2 O. Distilled water, MeOH, acetone, PC, and solutes were purchased from FUJIFILM Wako Corp. and used as received.Table 1 shows the solubility s and critical concentration c* of Fe(ClO 4 ) 2 /Fe(ClO 4 ) 3 in the four solvents.c* is dened as the concentration at which one Fe ion is dissolved per six solvent molecules.At c = c*, all solvent molecules are on average coordinated with Fe ions.

Total resistance
The total resistance R tot of the electrolyte was measured in a two-pole cell at 298 K. 24 The electrodes were produced from a 220 mm graphite sheet (PREMA-FOIL, TOYO TANSO).The electrode distance d and area s are 1.0 cm and 0.42 cm 2 , respectively.The voltage drop V was measured against the current I (I # 0.4 mA) with a multimeter.I was changed in a stepwise manner at intervals of several minutes.V was stable and no change over time was observed.The slope of the I-V plot corresponds to R tot .

Electrochemical impedance spectroscopy
R ct and R dif were evaluated in the same cell with the same electrodes.d and s are 1.0 cm and 0.42 cm 2 , respectively.Electrochemical impedance spectroscopy (EIS) was performed at 298 K with use of a potentiostat (Vertex.one.EIS, Ivium Technologies).The frequency range was from 50 mHz to 100 kHz, and the amplitude was 10 mV.V was stable and no change over time was observed.It was conrmed that almost identical EIS data were obtained through multiple measurements.
The EIS data were analyzed with a Randles equivalent circuit, 23 which consists of R s , R ct , double layer capacitance C d , and Warburg impedance Z u .Z u is expressed as ), where A W and u are the Warburg coefficient and angular velocity, respectively.It was difficult to evaluate the magnitude of R dif from A W even though Z u describes the diffusion process of the reactants/products.In the present study, we tentatively evaluate the R dif values by subtraction of R s and R ct from R tot .We conrmed a positive correlation between A W and R dif (= R tot − R s − R ct ), which strongly supports the correctness of our evaluation method of R dif (vide infra).

Total resistance
Fig. 1 shows examples of the I-V plot of several solutions containing dissolved Fe 2+ /Fe 3+ at 298 K: (a) water, (b) MeOH, (c) acetone, and (d) PC.For all solutions, V increases in proportion to I. R tot was evaluated from the slope of the plots, as indicated by the straight lines.The obtained R tot values are listed in Table Paper RSC Advances 2. In (a) the aqueous solution, R tot decreases from 65.0 U at 0.5 M to 28.5 U at 1.5 M, and then increases to 37.9 U at 2.5 M. The decrease in R tot in the low c region is due to the increase in the number (f c) of charge carriers, such as Fe 2+ and Fe 3+ .This behavior is consistent with the literature. 6Similar local minima structures in the c-R tot plot are also observed for the MeOH, acetone and PC solutions (Table 2).
Let us estimate the maximum value of ZT in the aqueous electrolyte at 300 K with the use of R tot shown in Table 2.The maximum value of s ¼ d SR tpt is 86.0 mS cm −1 at c = 2.0 M. Kim et al. 6 reported c-dependence of a and k in an aqueous solution containing Fe(ClO 4 ) 2 /Fe(ClO 4 ) 3 .From the extrapolation of the reported data, we evaluated a = 1.76 mV K −1 and k = 0.4 W K −1 m −1 at 2.0 M.Then, we obtained ZT = 0.020 at 2.0 M. The ZT value is smaller than the value (= 0.036 (ref.6) at 0.8 M) reported by Kim et al., 6 reecting the smaller s obtained in the present experiment.We note that effective s of a LTE is inuenced by the microscopic structure of the electrodes as well as the convection of the electrolyte.Table 2 Total resistance R tot , solution resistance R s , charge-transfer resistance R ct , diffusion resistance R dif , and Warburg coefficient A W in solvents containing c M Fe(ClO 4 ) 2 and c M Fe(ClO 4 ) 3 .R dif was evaluated by subtraction of R s and  straight line with the horizontal axis corresponds to R s + R ct − 2A W 2 C d .Similar behaviors are also observed in the other solutions.The solid curves in Fig. 1 are the results of least-squares ts with a Randles equivalent circuit composed of R s , R ct , C d , and Z u .The Randles equivalent circuit well reproduces the observed complex impedance.Thus, we obtained R s , R ct , C d , and A W .We further evaluate R dif (= R tot − R s − R ct ) with the use of R tot .The obtained R s , R ct , R dif , and A W values are listed in Table 2.

Electrochemical impedance spectroscopy
A W is expected to have a strong correlation with R dif because Z u [= A W (u −1/2 − iu −1/2 )] describes the diffusion process of the reactants/products.We calculated the correlation coefficient X between A W and R dif (= R tot − R s − R ct ) for each solution system; X = 0.976 for water, 0.995 for MeOH, 0.980 for acetone, and 0.988 for PC.The positive correlation (X $ 0.976) between A W and R dif strongly supports the correctness of our evaluation method of R dif .In this sense, reducing R s is effective to reduce R tot in organic solution.Shortening d is especially effective because k (z 0.2 W K −1 m −1 ) of an organic solvent is much smaller than k (= 0.6 W K −1 m −1 ) of water.Reecting the small k in organic solvent, a sufficient DT is expected between the electrodes, even in the cell with smaller d. .As a result, R ct becomes constant at sufficiently large c.

Concentration dependence of R s
Now, let us discuss the solution parameters that determine R s . 23where F, z j , u j , and C j are the Faraday constant, charge number, mobility, and molar concentration of the j-th ion, respectively.By substituting u j ¼ z j e 6phr j , we obtain R s À1 ¼ sFec 6pdh  (solid curves) against c.We note that there is only one tting parameter (C s ) to adjust the magnitude but no parameter to adjust the shape.Nevertheless, the curve reproduces the observed R s −1 well, except for (c) acetone solution.In Table 3, we listed C s .Except for the acetone solution, the solvent dependence of C s is rather small, falling between 0.104 mPa s M −1 U −1 and 0.183 mPa s M −1 U −1 .This is probably because the r value does not change greatly depending on the solvent.
In (c) acetone solution, the shape of the c-R s −1 plot (open circles) is qualitatively different from the shape of the empirical formula (solid curve).In the region of c $ 0.5, the empirical formula results decrease steeply while the observed R s −1 decreases slowly.If C s is set to ∼0.1, the agreement between the calculated and observed values is improved in the region of c $ 0.5 even though the calculated value is much larger in the region of c ∼ 0.3.This implies that an additional factor, e.g., the repulsive interaction between Fe ions, suppressed R s in the region of c ∼ 0.3.

Concentration dependence of R dif
Finally, let us consider the relationship between R dif −1 and h.formula (solid curves) against c.We note that there is only one tting parameter (C dif ) to adjust the magnitude.Nevertheless, the curve reproduces the observed R dif −1 well.In Table 3, we list the C dif values.The solvent dependence of C dif is rather small, falling between 0.053 mPa 1/2 s 1/2 M −1 U −1 and 0.166 mPa 1/2 s 1/2 M −1 U −1 .This is probably because the r value does not change greatly depending on the solvent.

Conclusions
In conclusion, we investigated the c-dependence of R s , R ct , and R dif in dissolved-Fe 2+ /Fe 3+ -containing aqueous, MeOH, acetone, and PC solutions.We found that the c-dependence of R s and R dif is well reproduced by the empirical formulas R s À1 ¼ C s c hðcÞ and  R dif À1 ¼ C dif c hðcÞ 1=2 .We further found that the magnitudes of C s and C dif are nearly independent of the solvent, suggesting that h is one of the signicant solution parameters that determine R s and R dif .Our ndings suggest that s of the electrolyte solution can be increased through reducing h.

Fig. 2
Fig. 2 shows examples of the Cole-Cole plots of complex impedance for several solutions containing dissolved Fe 2+ /Fe 3+ at 298 K: (a) water, (b) MeOH, (c) acetone, and (d) PC.The Cole-Cole plot of 0.1 M MeOH solution (Fig. 2(b)) shows a prototypical shape.The plot shows a semicircle on the le side and

Fig. 1
Fig. 1 Voltage V against current I for several solutions containing dissolved Fe 2+ /Fe 3+ at 298 K: (a) water, (b) MeOH, (c) acetone, and (d) PC.Data for three typical solute concentrations are shown for each solution: below the resistance minimum (blue), near the resistance minimum (green), and above the resistance minimum (red).The straight lines are the results of least-squares fits.

Fig. 2
Fig. 2 Cole-Cole plots of complex impedance for several solutions containing dissolved Fe 2+ /Fe 3+ at 298 K: (a) water, (b) MeOH, (c) acetone, and (d) PC.The solid curves are the results of least-squares fits with a Randles equivalent circuit composed of R s , R ct , C d , and Z u (see text).

Fig. 3
Fig. 3 shows the c-dependence of (a) R s −1 , (b) R ct −1 , (c) R dif −1 , and (d) R tot −1 in several solutions containing dissolved Fe 2+ /Fe 3+ at 298 K.For the convenience of explanation, the horizontal axis is normalized by the critical concentration c* of each solvent.At c = c*, all solvent molecules are on average coordinated with Fe ions.

and R dif − 1 .
In the small c c* region, R s −1 increases linearly with c c* as indicated by the straight lines in Fig. 3(a).The increase in R s −1 is due to the increase in the number (f c) of charge carriers, such as Fe 2+ and Fe 3+ .Upon further increasing c c* beyond ∼0.3,R s −1 begins to decrease with c c* .Similarly, in the small c c* region, R ct −1 and R dif −1 increase linearly with c c* as indicated by the straight lines in Fig. 3(b) and (c), respectively.The increase in R ct −1 and R dif −1 is due to the increase (f c) of reactant/product concentration, i.e., Fe 2+ /Fe 3+ .Upon further increasing c c* beyond ∼0.5, R ct −1 and R dif −1 begin to saturate.The saturation of R ct −1 can be ascribed to the nite reaction number (N reaction ) per unit time at the electrode surface.The redox reaction cannot keep up with the supply of reactants when the number (N reactant f c) of reaching reactants per unit time exceeds N reaction .In such a region, N reaction becomes the rate-determining factor for the charge-transfer current J ct , and hence, R ct −1 C Fe 2+ = C Fe 3+ = c in the present solutions.By assuming P j z j 2 r j is independent of c in each solution, we obtain the simple relation R s À1 fC s c h , where C s is a constant.The top panels of Fig. 4(a)-(d) show h of each solution against c.The h values were evaluated at 298 K using a sine-wave vibro viscometer (SV-10; A&D Company Limited).In all solutions, h increases nonlinearly with c.The solid curves are the results of least-

Fig. 3 1 ,
Fig. 3 Solute concentration (c) dependence of (a) R s −1 , (b) R ct −1 , (c) R dif −1 , and (d) R tot −1 in several solutions containing dissolved Fe 2+ /Fe 3+ at 298 K.The horizontal axis is normalized by the critical concentration c* of each solvent.The straight lines in (a), (b), and (c) represent the linear relation with c c* .

R dif − 1
is proportional to the diffusion current J dif , which is expressed as J dif fD dC dx .Replacing the differential with the difference, we get J dif f DDC Dx ,23 where Dx and DC are the diffusion length and concentration difference between electrode surface and bulk solution, respectively.In one-dimensional diffusion, Dx is expressed as Dx ¼ ffiffiffiffiffiffiffi ffi 2Dt p fD 1=2 , where t is the elapsed time.Then, J dif is proportional to cD 1/2 because DC f c.From the Stokes-Einstein equation, we obtainD ¼ k B T 6phr .Finally, we obtain an empirical relationR dif À1 ¼ C dif c hðcÞ 1=2, where C dif is a constant.We can calculate the empirical formula with use of the quadratic function h(c).The bottom panels of Fig.4(a)-(d) show comparisons between the observed R dif −1 (open circles) and empirical

Fig. 4 1 , 1 , 1 ,
Fig. 4 (a) Viscosity h (top), R s −1 (middle), R dif −1 (bottom) in aqueous solutions containing dissolved Fe 2+ /Fe 3+ at 298 K against solute concentration c.(b) h, R s −1 , and R dif −1 in MeOH solutions containing dissolved Fe 2+ /Fe 3+ at 298 K against c.(c) h, R s −1 , and R dif −1 in acetone solutions containing dissolved Fe 2+ /Fe 3+ at 298 K against c.(d) h, R s −1 , and R dif −1 in PC solutions containing dissolved Fe 2+ /Fe 3+ at 298 K against c.The solid curves in the upper panels are the results of leastsquares fits with a quadratic function.The solid curves in the middle panels are the results of least-squares fits with the c hðcÞ function.The solid curves in the bottom panels are the results of least-squares fits with the c hðcÞ 1=2 function.
23Among them, R s is derived from the balance between the electric force (=jzjeE ef ; jzj, e, and E ef are the charge number,

Table 1
Solubility s and critical concentration c* of Fe(ClO 4 ) 2 -$6.0H 2 O/Fe(ClO 4 ) 3 $7.1H 2 O in several solvents at 298 K. c* is defined as the concentration at which one Fe ion is dissolved per six solvent molecules.MeOH and PC represent methanol and propylene carbonate, respectively © 2024 The Author(s).Published by the Royal Society of Chemistry RSC Adv., 2024, 14, 6292-6297 | 6293

Table 3
Coefficients (C s and C dif ) of empirical formulas, R s À1 ¼ C s c hðcÞ and R dif À1 ¼ C dif c hðcÞ 1=2 , determined by least-squares fits with the observed data.MeOH and PC represent methanol and propylene carbonate, respectively Solvent C s (mPa s M −1 U −1 ) C dif (mPa 1/2 s 1/2 M −1 U −1 )